Research On The Rosenbloom-Tsfasman Metric Of Error-correcting Codes Over Finite Rings

Let C (?) F_qⁿ be a linear code. Let n = ms, m and s being positive integers. Then the elements of C can be viewed as matrices P=(P_ij), 1≤i≤m,1≤j≤s, with entries in F_q . The Rosenbloom-Tsfasman weight (RT, orρ, in short) of P is defined as:And the RT metric of P and Q is defined as p(P,Q) = w_N(P-Q). For s = 1 , the RT metric is just the usual Hamming metric. We can see that the RT metric is a non-Hamming metric and is a generalization of the usual Hamming metric, so the study of it is very significant from both a theoretical and a practical viewpoint. There has been a recent growth of interest in codes with respect to this RT metric.The weight distribution of codes, in a classical coding theory, is an important investigation field. Whether in fields or in rings, using weight distribution of linear codes to explore the weight distribution of their dual codes is very significant.In this paper, the distribution of linear codes with respect to the RT metric and the RT weight over finite rings is explored. Some kinds of weight enumerators with respect to the RT metric over M_n×s(F₂ +uF)₂ and M_n×s(Z₄) rings are defined, and the corresponding Mac Williams identities are proven too. Also, a definition of the generalized RT weights of binary codes is obtained, basic properties of the generalized weights are derived, and the values of these weights for well-known classes of codes are determined too. At last, we summarize the paper, and put forword some problems which will be some developmental direction in the future.